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The $\alpha$-$T_3$ system undergoes a topological phase transition(TPT)between two distinct quantum spin-Hall phases across $\alpha=0.5$ when thespin-orbit interaction of Kane-Mele type is taken into consideration. As ahallmark of such a TPT, we find that the Berry curvature and the orbitalmagnetic moment change their respective signs across the TPT. The trails of theTPT found in another physical observable e.g. the orbital magnetization(OM) areunderstood in terms of valley and spin physics. The valley-resolved OM(VROM)and the spin-resolved OM(SROM) exhibit interesting characteristics related tothe valley and the spin Chern number when the chemical potential is tuned inthe forbidden gap(s) of the energy spectrum. In particular, we find that theslope of the VROM vs the chemical potential in the forbidden gap changes itssign abruptly across the TPT which is also consistent with the correspondingchange in the valley Chern number. Moreover, the slope of the SROM demonstratesa sudden jump by one unit of $e/h$, (where $e$ is the electronic charge and $h$is the Planck's constant), across the TPT which is also in agreement with thecorresponding change in the spin Chern number.

Orbital magnetization senses topological phase transition in spin-orbit coupled $\alpha$-$T_3$ system